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Sequentially Compactness

  1. Sep 16, 2008 #1
    Suppose that S is a nonempty set of real numbers that is not Sequentially compact. Prove that either (i) there is an unbounded seqeunce in S or (ii) there is a sequence in S that converges to a point x0 that is not in S.

    I am having trouble with this it not being sequentially compact is screwing me up, I don't know how to prove it.
  2. jcsd
  3. Sep 16, 2008 #2


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    Well, there are two ways you can proceed:
    (1) Suppose that S is bounded, and prove that there is a sequence in S that converges to something outside of S, or
    (2) suppose that every convergent sequence in S has a limit in S, and prove that S cannot be bounded.

    Have you tried either way? If so, what kind of problems did you run into?
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